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Sequences, pairwise different sums

Source: Kürschák 2001, problem 2

July 13, 2014
combinatorics unsolvedcombinatorics

Problem Statement

Let k≥3k\ge 3 be an integer. Prove that if n>(k3)n>\binom k3, then for any 3n3n pairwise different real numbers ai,bi,cia_i,b_i,c_i (1≤i≤n1\le i\le n), among the numbers ai+bia_i+b_i, ai+cia_i+c_i, bi+cib_i+c_i, one can find at least k+1k+1 pairwise different numbers. Show that this is not always the case when n=(k3)n=\binom k3.
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